Idaho Skies

March 2008

Vol. 5 No. 2

Idaho Skies is a column for beginning amateur astronomers and those interested in astronomy. Suggestions about the column are gladly accepted by the columnist, at paul.verhage@boiseschools.org

This month look for the star, Capella, the lucida of Auriga, The Charioteer. Called Alpha Aurigae by astronomers, Capella is located high in the northwest and to the left of the bowl of the Big Dipper in March. If you were born in 1966, then Capella is your birthday star this year. That’s because the light of Capella that you see tonight left the star 42 years ago. The name Capella comes from the Latin for little she-goat. This name is in reference to the fact that Auriga is seen holding young goats in depictions of the constellation. The goats are the three stars forming a narrow triangle very close and just south of the star Capella.

Capella, the sixth brightest star in the heavens, is really two large yellow giant stars in orbit around each other. Their distance of roughly 60 millions miles apart makes them too close together at their 42 light-years to see as separate stars except under the best circumstances and telescopes. The yellow stars of Capella are over twice as heavy as the sun but millions of years ago, were probably as white and bright as Sirius is now. Today they’re old enough that one is fusing helium in its core and the other is contracting and will begin fusing helium in the near term. With their larger masses, the stars are ten times larger that the sun and 50 and 80 times brighter.

March 1 – 7

We can Jupiter again! On the morning of the 2^nd at 6:00 AM, Jupiter appears as the bright star to the moon’s left. The celestial pair is low in the east. Their distance apart of 7 degrees should just fit within the width of your binoculars field of view. Tomorrow morning Jupiter appears the same distance from the moon but to its right. Jupiter is one of the best planets for the telescope. Its brightness, size, and ever changing moons put on a great telescopic show.

The moon is new on the 7^th. If you could see it, you’d see the moon pass just above the sun from our perspective. Somewhere, in space way above the North Pole, there’s a total solar eclipse today.

March 8 – 14

Daylight Saving Time ends on the morning of the 9^th. Be sure to see your clocks ahead by one hour before you go to bed on the 8^th.

The star cluster M-35 in Gemini is easy to find on the night of the 10^th because of Mars. Mars, the bright yellow-orange star high overhead in the southwest, is less than two degrees above the cluster. For reference, an angle of two degrees is less than one-third of your binocular’s field of view. If you place Mars in the center of your binoculars, you’ll notice a faint fuzzy spot most of the way towards the bottom. The cluster is more visible if you move brighter Mars out of sight. How many stars in this cluster can you count?

Also on the 10^th, the moon is at perigee, or its closest to earth. The distance between the centers of the earth and the moon is 227,607 miles today. That’s a significant distance compared to the distances we travel on earth. However, compared to the rest of the solar system, it’s peanuts. Why even the distance across the sun’s equator is nearly four times greater.

Half way up in the west on the evening of the 12^th, you’ll see the moon near the Pleiades. Their distance apart is five degrees, or 10 lunar diameters. To the moon’s left is the sparser star cluster, the Hyades. Both star clusters are nice binocular objects.

The moon reaches the first quarter phase on the 14^th. You’ll see rusty Mars three lunar diameters below the moon and the star cluster M-35 four lunar diameters farther below. The entire trio will fit well within your binocular’s field of view as illustrated below.

March 15 – 21

The Ulysses spacecraft finishes its third pass over the sun’s North Pole on the 15^th. You’ve never heard of Ulysses? That’s not too surprising, not may people have. Launched 18 years ago, Ulysses is unique in that it orbits the solar system in a high inclination orbit. Virtually every spacecraft orbits the earth or sun in an orbit that lies in the same plane as the earth’s orbit around the sun. This means spacecraft designed to study the sun are limited to observing near its equator. In order to observe to sun’s polar regions, a spacecraft in an orbit with a high inclination is needed. This is what the Ulysses spacecraft was launched to do. The 800 pound robotic explorer loops out to the orbit of Jupiter and back in again over the solar poles. It’s suite of instruments study the sun’s gamma and x-ray radiation, magnetic field, and solar wind. In addition, Ulysses reports back on the interstellar dust and gas reaching the solar system from high latitudes. You can read about this little know, but unique mission at its website, http://ulysses.jpl.nasa.gov

Fifty years ago on the 17^th, the United States launched the Vanguard satellite into earth orbit. Three months earlier, the original Vanguard booster exploded at lift-off in front of millions on television. The second was successful and Vanguard became our second satellite in earth orbit and the fourth in the history of the space age. The Vanguard booster was so good and Vanguard so light that the spacecraft is still in orbit today. Vanguard is a six inch aluminum sphere with six antennas and six tiny solar cells. Its two radio transmitters enabled tracking of the satellite and a determination of the amount of free electrons in space and the atmosphere between the satellite and ground station. Because of atmospheric drag even at Vanguard’s high altitude (it climbs to over 2,000 miles high), the satellite will return to earth’s atmospheric in 240 years. Perhaps we can recover the world’s oldest satellite before that happens. One of Vanguard’s most interesting discoveries is that the world is not round like a sphere, but is more pear shaped.

On the evening of the 17^th, you can use the moon to find the Beehive star cluster. To the moon’s right and 5-1/2 degrees away you’ll see a star cluster larger than the moon’s diameter (as seen from earth). The angular distance between the moon and the Beehive is nearly the distance across your binoculars field of view. So once you have the moon in sight, turn your binoculars to the right until the moon is close to the left edge of the field of view. You’re looking for a scattering of stars on the right that will remind you of a swarm of bees.

On the 18^th, the moon leads you to Regulus and Saturn. From the moon to Saturn spans an angle of 5-1/2 degrees, or less than the field of view of your binoculars. Regulus is the lucida, or brightest star of Leo the Lion. It’s at a distance of 77 light years. Saturn will appear star-like and yellow in your binoculars. But if you have a telescope with a magnification of at least 25 power, then you can see its rings and brightest satellite, Titan.

The Vernal Equinox occurs on the 19^th at 10:48 PM. That’s the beginning of Spring in the northern hemisphere. Today, the day is 12 hours and 5 minutes long. If you could see the earth’s equator projected into space, the sun would be sitting on it today. From now until the first day of autumn, the sun will be north of the earth’s equator and we’ll enjoy days that are longer than the night.

The moon is full on the 21^st. I hope you weren’t planning to look for faint galaxies and nebulas tonight. The full moon in March is often called the Sap Moon.

March 22 – 30

Beginning on the evening of the 23^rd, the Zodiacal Light is visible in the west after it gets dark. The Zodiacal Light appears as a faint triangle of light reaching as high as half way up in the sky. The light is actually sunlight, but reflected from dust orbiting between earth and the sun. The source of this dust is asteroid collisions and comet tails. The dust is being constantly replenished because asteroid collisions and comets are ongoing events in our solar system. The Zodiacal Light will remain visible for the next two weeks before evening moonlight begins interfering.

On the 26^th, the moon reaches the apogee of its orbit. Non-circular (elliptical) orbits like the moon’s have a closest point and farthest point. Astronomers call the farthest point of an orbit the apogee if it’s an orbit around the earth (that’s were the “gee” in apogee comes from). This month’s lunar apogee is 251,712 miles from the center of the earth.

Do you want someone to point out the star Antares to you? Then get up early on the 27^th to see the moon do just that. Antares, the red giant lucida of Scorpius the Scorpion will be 1-1/2 degrees, or three lunar diameters, to the upper right of moon.

The moon reaches last quarter phase on the 29^th. Now that the weather is finally getting comfortable again, tonight would make a great time to go moon watching. Be sure to focus your attention on the long shadows at the lunar terminator, or straight-edged boundary between day and night. That’s where you’ll see the most detail in the lunar terrain.

This Month’s Topic

Pi

Pi is probably the most famous number in history. Defined as the ratio of the circumference of a circle to its diameter, pi equals approximately 3.14159. The Greek letter pi (π) is equivalent to our letter “P” and was chosen to represent the ratio for a circle’s circumference to its diameter because it‘s the first letter in the Greek word for perimeter. While we pronounce the letter like “pie”, the Greeks would have pronounced it more like “pee”.

There’s a funny thing about the number pi however, it cannot be written. If you were to take the circumference of a circle and divide it by its diameter, you would be writing out the result forever. That’s because the number of digits in the number pi is infinitely long. In mathematics, numbers like this are called irrational. Being irrational not only means the number never ends, but that there’s also no repeating pattern in the number or anyway to write a finite fraction that represents the number. As an example, the number 12121212…. may go on forever, its not irrational because the number is a repeating pattern of 12. This infinitely long number can be written as the simple fraction, 12/99. Pi on the other hand is irrational. Therefore, no matter how many of the digits in pi you write out, the pattern of digits will never begin repeating nor is there a nice fraction that you can use in its place.

There are some number irrational numbers that can be calculated with a simple equation. However, pi is not one of these. That means pi is also a transcendental number. That doesn’t mean you can’t calculate pi with an equation, indeed there are many equations for calculating pi. It means however that the equation itself is also infinitely long (these formulas are called infinite series). Here’s one example of an infinite series formula you can use to calculate pi.

Pi = + 4/1 – 4/3 + 4/5 – 4/7 + 4/9 – 4/11 + 4/13 – 4/15 …………

In this formula, the numerator (top number in a fraction) is always 4 and the denominator (bottom number in a fraction) is always an increasing odd number. Each fraction in the infinite series alternates between addition and subtraction. Mathematically, you could say the fractions alternate between adding a positive or negative value of the fractions. A spreadsheet can easily calculate pi using this method using the cell formulas shown below.

After entering these equations, copy and paste rows 2 and 3 as many times you want the spreadsheet to calculate pi. After 100 calculations, the spreadsheet generates the following chart of pi’s calculated value.

You can see that this infinite series is not very good at calculating the value of pi. It requires far too many reiterations to narrow the value of pi to its precise value. In other words, the series doesn’t converge on pi very quickly. Even after 100 iterations, the calculated value of pi alternates between 3.13 and 3.15.

The ancients, who evaluated the value of pi geometrically, determined it was between 223/71 and 22/7. That’s where things stood until the 1400’s when more advanced mathematics, including calculus, was brought to bear on the value of pi. Calculus allows the more effective development and evaluation of infinite series. Newton, one of the developers of calculus, developed an infinite series for calculating pi, which let him determine its value to 15 decimal places in the mid 17^th century.

One sad calculation for the value of pi occurred at the end of the 19^th century. Mathematician William Shanks spent 15 years calculating the first 707 digits in pi. Later evaluations showed that he had made a mistake in his work so only the first 527 digits were correct. Once you make one mistake in an infinite series, the rest of the calculations are a wasted effort.

Today computers calculate the value of pi. Super computers have determined pi’s value to a precision of 100 trillion digits. This level of precision isn’t required for most uses of pi; it could calculate the circumference of the universe to a precision smaller than a hydrogen atom.

Did you know some people make a game out of memorizing the value of pi? A pi poem, or piem is a poem where the letters in each word are the same as the digits in pi. For example, 3.14, the first three digits of pi could be “coded into” the piem, “cut a page”. Using this method, people have memorized the first 3,834 digits of pi.

Pi appears in what mathematicians considered the most beautiful equation in mathematics. However, to describe it, you need to be familiar with two other numbers, i and e. You’re probably already familiar with the number i. It represents the value of the square root of -1. There’s no positive or negative number that when multiplied by itself equals -1. Therefore, we can’t actually write a number for i like we can for pi. Mathematicians call numbers like i imaginary numbers. The number e shows up in numbers that increase in size proportionally to their size. For example, if an equation doubles the size of a number twice as fast as the same equation would double a number half as large, then the number e appears in it. One place you see this in the real world is in bank loans. For example, the amount of interest you pay on a load of $100 is twice as much interest you’ll pay on a load of $50. The number e is like pi in that it is irrational and its value to four digits is 2.718.

Now here’s Euler’s Identity, the most beautiful equation in mathematics.

e^iπ = -1

It amazingly relates irrational, imaginary, and real numbers. Who would ever have thought an irrational number to the power of a imaginary and second irrational number would equal a negative real number?

This Month’s Sources

The Royal Society of Canada, Observer’s Handbook 2008

Baalke, Ron, Space Calendar, January 13, 2008, <www.jpl.nasa.gov/calendar/>

Kaler, James, Stars: Capella, 1 February 2008, <www.astro.uiuc.edu/~kaler/sow/>

Night Sky Explorer (software)

NASA, Ulysses, December 7, 2007, January 13, 2008, <http://ulysses.jpl.nasa.gov/>

Wikipedia, Vanguard 1, November 29, 2007, January 13, 2008, <http://en.wikipedia.org/wiki/Vanguard_1>

Wikipedia, Pi, 28 January 2008, 29 January 2008, <http://en.wikipedia.org/wiki/Pi>

Dark Skies and Bright Stars,

Your Interstellar Guide